It's a bit of effort the first time but inexpensive and easy after that...
You need to get a UPS account. Just go to this website::
Now scroll down to "Open a free account", click on that and give them the required info. I can't go through the whole process because they recognize me from cookies and I don't need a second account. They'll give you an account number and along with it you get a pretty substantial discount.
Pack the ammo up. It should be in double thickness cardboard which I never have so I hot glue a cut up box inside another. Tape it up then weigh it. If you don't have an appropriate scale just get on a bathroom scale with the ammo then set the ammo on a shelf and do the arithmetic. A nice lady agent at a UPS center suggested this to me and added "The only time we question the weight is if you claim it weighs two pounds and the driver can't pick it up by himself"
Log on to your account, fill in the addressee info, weight, dimensions, etc. and print the label. Stick an ORM-D label on the box. If you don't have an ORM-D label you can just draw one on the box with a magic marker. Plenty of pictures of them online.
You can pay UPS to make a pickup at your place, you can take it to a UPS Customer Center (they always have unusual hours), or if you know a local business that has daily pickup they might let you drop it there. I used my local drug store for years before I moved to Florida. You can also just hand it to a UPS driver if you see one making a delivery. DO NOT take it to The UPS Store under any circumstances. That's a whole new bag of snakes.
It sounds like a hassle but the savings of having a UPS account are very significant and the process is simple once you've done it a couple times. In addition to occasionally shipping some ammo we also ship Christmas gifts, birthday presents and all manner of stuff simply because it's the cheapest way.
I shipped about a dozen boxes with three bricks of 22 LR in each one, weight was 12 pounds. The prices ranged from $12.25 for Florida to Georgia and $18.77 from Florida to Washington state. The UPS published rate for 12 lbs from FL to WA is $41.10 so they knocked 54% off because I have a customer number.